On the Parameterized Complexity of Computing Balanced Partitions in Graphs

Abstract

A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters.For the Vertex Bisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that Vertex Bisection is W[1]-hard w.r.t. (k, c).Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of n O(d) . We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number. * An extended abstract of this article appeared at the 39th International Workshop on GraphTheoretic Concepts in Computer Science (WG 2013) [50]. The extended abstract contains results regarding Bisection and Vertex Bisection, while this article additionally provides full proof details as well as parameterized complexity analyses of Balanced Partitioning. This article is published in Theory of Computing Systems, available at link.springer.com.

The Complexity of Tree Partitioning

“…This also closes the door on the complexity of these problems on trees, as the simple case when the tree is a path is in P. (B) We prove that both Tree Partitioning and Balanced Tree Partitioning are W [1]-complete. This answers an open question in [25]. We observe that, for trees, the removal of k edges results in k + 1 components.…”

“…Bevern et al [25] also showed that Balanced Graph Partitioning is W [1]-hard on forests by a reduction from the Unary Bin Packing problem, which was shown to be W [1]-hard in [19]. We note that the disconnectedness of the forest is crucial to their reduction, as they represent each number x in an instance of Bin Packing as a separate path of x vertices.…”

“…Bevern et al [25] showed that the parameterized complexity of Balanced Graph Partitioning is W [1]-hard when parameterized by the combined parameters (k, µ), where k is (an upper bound on) the cut size, and µ is (an upper bound on) the number of resulting components after the cut. It was observed in [25], however, that the employed FPT -reduction yields graphs of unbounded treewidth, which motivated the authors to ask about the parameterized complexity of the problem for graphs of bounded treewidth, and in particular for trees. We answer their question by showing that the problem is W [1]-complete.…”

“…For Balanced Tree Partitioning, in contrast to Unary Bin Packing (and hence, to Balanced Graph Partitioning on forests), the difficulty is not in grouping the components into groups (bins) because enumerating all possible distributions of k + 1 components (resulting from cutting k edges) into b ≤ k + 1 groups can be done in FPT -time; the difficulty, however, stems from not knowing which tree edges to cut. The FPT -reduction we use to show the W [1]-hardness is substantially different from both of those in [19,25], even though we use the idea of nonaveraging sets in our construction-a well-studied notion in the literature (e.g., see [4]), which was used for the W [1]-hardness result of Unary Bin Packing in [19].…”

Recent Advances in Graph Partitioning

“…On finite connected subgraphs of the two dimensional grid without holes, the bipartitioning problem can be solved optimally in O(n 4 ) time [FW11]. Recent work by Bevern et al [vBFSS13] looks at the parameterized complexity for computing balanced partitions in graphs.…”